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# 446r - MATH 446 OR 481 SAUER SPRING 2010 Review Problems 27...

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MATH 446 / OR 481 SAUER SPRING 2010 Review Problems 27 April 2010 1. Describe the most efficient method you can find (in the sense of the minimum number of multiplications needed) for evaluating the polynomial P ( x ) = a 3 x 3 + a 7 x 7 + a 11 x 11 + a 15 x 15 + a 19 x 19 + a 23 x 23 . How many multiplications are needed for each input value of x ? 2. Find the IEEE double-precision machine representation of the decimal numbers (a) 15 (b) 1 / 8 (c) 0 . 8 (d) 0 . 7 3. (a) Convert the binary number 11 . 1 001 to base ten. (b) Convert 31 / 6 to binary. (c) If 31 / 6 is to be stored as a double precision number using IEEE rounding to nearest, find the exponent, and list the four rightmost bits in the stored number. 4. Calculate (a) (2 + (2 - 51 + 2 - 52 )) - 2 and (b) (2 + (2 - 51 + 2 - 52 + 2 - 53 )) - 2 and express the answers in terms of machine epsilon. 5. Calculate both roots of x 2 + 2 60 x = 1 to 3 correct decimal digits. 6. (a) Solve 36 x 2 + 36 x = 7 for the degree 1 term and set up the resulting fixed point iteration. (b) Show that 1 / 6 and - 7 / 6 are fixed points. (c) Which of the fixed points are locally con- vergent? (d) If FPI is run with initial guess x = 1 , what will happen? (e) Same as (d), but for initial guess x = 2 . 7. (a) Show that x = 4 / 3 and x = 5 / 3 are fixed points of the equation x = ( x - 1) 2 + 11 / 9 . (b) Which fixed point will attract initial guesses under FPI? (c) If FPI is run with initial guess x = 1 , what will happen? (d) Same as (c), but for initial guess x = 2 . 8. Develop a method for computing the fifth root of a number a to several correct digits. It should require only elementary operations like multiplications, divisions, additions, etc. Discuss the convergence properties of your algorithm. 9. Assume that Newton’s method is applied to find the roots of f ( x ) = x 4 + 2 x 3 - 2 x - 1 , and assume that after 4 steps you are within e 4 = 0 . 0001 of the root. Estimate the total number of steps required to calculate the following roots within 50 correct decimal places: (a) r = - 1 (b) r = 1 . 10. For f ( x ) = x 4 - (7 / 2) x 3 + (15 / 4) x 2 - (13 / 8) x + 1 / 4 , does Newton’s method converge faster or slower than the bisection method to x = 1 / 2 ? What about to x = 2 ? 11. Let f ( x ) = x 3 + 2 x - 3 .

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446r - MATH 446 OR 481 SAUER SPRING 2010 Review Problems 27...

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